Just off the top of my head, I can't think of anything quite as stripped-down as that, on the jazz side of things, but perhaps you might like Diana Krall's 'All for You - a tribute to the Nat King Cole Trio' album, even though it's more instrumented.
youtube.com/results?search_query=diana+krall+all+for+you
Kinga Hemming can be quite laid-back: youtube.com/watch?v=NOSf67TPJtk
Claire Martin, too: youtu.be/RQyl2mhnuhM
and Stacey Kent youtu.be/vAyBJ59z4LY
On the Instrumental jazz side of things, Coleman Hawkins & Ben Webster ain't bad:
youtube.com/watch?v=hi8NtRBfrAc
If you don't mind older recordings, Chet Baker deserves a mention:
youtube.com/watch?v=jvXywhJpOKs
Apologies that these aren't quite as 'stripped' as you were looking for, but nothing, to that degree, springs to mind at the moment.
If I think of anything else, I'll repost.
Aside from Jazz:
Shelby Lynne's album 'Just a Little Lovin'' is a pretty-stripped-back nod to the career of Dusty Springfield. Here are a couple of tracks from the album:
youtube.com/watch?v=Ye6Vw6_aR18
youtube.com/watch?v=fwf844beG5E
and her cover of Tony Joe White's 'Rainy Night in Georgia' is quite restrained, too:
youtube.com/watch?v=wBKlfmdFmZo
Ottmar Lienert plays decent acoustic guitar without too much additional instrumentation:
youtube.com/watch?v=oDWNp72vKQ8
Quadron (prior to selling-out to the music industry) did some pleasantly low-key performances:
youtube.com/watch?v=ILPNhmg3geM
As I'm typing this message, I'm listening to this: youtube.com/watch?v=v6R9H84koxs